2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper for exam preparation. Question paper for 2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper, Exam Question papers 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2012 university in india question papers. SiteMap
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2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper

University Question Papers
2005 Indian Statistical Institute M.Sc Mathematics General Topology University Question paper

Attempt any five questions. All questions carry equal marks. Any result
proved in the class may be cited and used without proof.

1. a) Let X be compact and Hausdorff, A ( X be closed. Show that X/A
is homeomorphic to the one-point compactification of X - A.

b) Describe explicitly the quotient topology on the quotient group
IR/|Q, IR being the real line, |Q the set of rationals, treated as a subgroup
of the group (IR, +).

2. a) Prove that GL(n, C) is path connected (hint; use the polynomial
p(z) = det((1 - z)I + zA) for A 2 GL(n, C)).

b) Prove that any discrete subgroup of S1 must necessarily be finite
cyclic.

3. a) Let X be any space. Show that CX, the cone over X is contractible.

b) Show that Sn-1 is a deformation retract of Sn - {N, S},N and S being the north and south poles of Sn respectively.

4. Let f, g : X ! Sn be continuous maps with f(x) 6= -g(x) 8 x 2 X.
Prove that f ' g.

5. Let X be a space. Then show that X is path connected if and only if
all constant maps: X ! X are homotopic to each other.

6. Let R_ : S1 ! S1 be a rotation by angle _. Show that R_ is homotopic
to the identity map: S1 ! S1.

7. Let G be a connected group, H a discrete normal subgroup. Prove that
H _ Z(G), the centre of G.



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